Having the right gear for your climbing style and level of fitness/coordination is most important.
Second is practicing with your gear in the offseason so that you’re cool under the pressure of running late, bumping deer, mad at a missed shot and not paying attention, get a call from home and have to rush, etc...
You want the movements you’re doing in the tree to be second nature. And you want protection that matches you.
I feel comfortable wearing a lineman’s only while climbing. But I do not let my linesman’s belt go loose without both hands around tree. If I am going to do anything that requires this, I’ll hook my tether up. Will also always hook tether up to go over a branch. I also make sure to pull myself into the tree as close as possible when I’m tying on a step or stick, or any other motion that requires both hands.
Like red said, climbing comes with inherent risk. If you have good ropes and climbing equipment, are practiced and skilled in their use, and focus on the task at hand, you’ll eliminate the lions share of the risk...